I flew the Ki-61 a lot a couple of tours ago. Here's what I remember about it.
(a) its a bobsled. Fly it like one. Never engage anything higher than it, happily engage anything lower than it.
(b) never attack anything below 10k altitude unless you've got clear sky and ample speed to get back up to 15k after you've dived down below the 10k floor.
(c) your cannons are in your nose, so convergence isn't an issue. Feel free to take close-in snapshots but don't try to hit anything with the nose guns beyond d400. Use the wing machine guns for the longer-range shots. The Ki-61 loves to shoot other planes' cockpits.
(d) the wings are fragile when hit. In fact, the whole airplane is fragile. Keep this in mind when strafing airfields and attacking bombers. In fact, don't attack bombers unless you can get in fast and get out fast.
(e) it can 0wn a lower Corsair and Hellcat. If either are above you, get them to turn until they run out of speed or are equal to you. Don't sacrifice altitude to chase them. It will take too long for you to climb back up.
(f) Forget about sneaking in low on an NOE raid. The clown-tailed Ki likes to be showy.
(g) leave the drop tanks at home. They'll just retard whatever bit of climb rate you have. Same with the bombs, unless you're on a milkrun to defend your base against a panzer, M8, and M3 attack.
(h) it can 0wn a Spitfire IX and all of the German planes in a turnfight. If you can avoid the head-ons by the Typhoons and LA-7s, you can kill them too after you turn around, assuming they don't run away once they realize they're in trouble.
(i) you can get a boatload of perk points in it with just a few kills. Surprisingly, the Ki-61 doesn't get used much, so most people don't know how to fight against it. Use their ignorance to your advantage by minding the information above.